Questions tagged [factorization]
For questions about factorization, the decomposition of mathematical objects (e.g. natural numbers, polynomials) into products of smaller objects (e.g. primes, lower degree polynomials).
291 questions
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$O(1)$ algorithm for factoring integers of the form $n=X (X^D+O(X^{D-1}))$
Factorization of integers of special forms are of both theoretical
interest and cryptographic implications.
Experimentally we found a seemingly "large" set of integers for which a divisor ...
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Closed form solutions of $x^D+y^D=n$ and $y^D-x^D=n$ assuming $y>x^{D/(D-1)}$
Let $x,y,X,Y,D>1$ be positive integers.
Let $X^D+Y^D=n$ and assume $Y>X^{D/(D-1)}$.
Conjecture 1 $Y=\lfloor n^{1/D}\rfloor$ and $X=(n-Y^D)^{1/D}$
Let $Y^D-X^D=n$ and assume $Y>X^{D/(D-1)}$.
...
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$O(1)$ algorithm for factoring integers of the form $p(p+O(\sqrt{p}))$?
Let $p,C$ be positive integers and assume $0 < C< \sqrt{p}$
Let $n=p (p+C) $ and assume $n$ is odd.
Conjecture 1 Given $n$ we can find $p,p+C$ with complexity computing two integer square roots ...
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Complexity of factoring $n=p q$ assuming $ 0 < q - p < p^C$
Let $p,q$ be positive integers and $ 0 < C < 1$ real.
Let $n=p q$ and assume $ 0 < q - p < p^C$
Q1 Is there $C$ such that given $n$ we can factor it (find $p,q$) in polynomial in $\log{p}$...
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On factoring integers of the form $n=(x^D+1)(y^D+1)$ and $n=(x^D+a_{D-2}x^{D-2}+\cdots+ a_0) (y^D+b_{D-2}y^{D-2}+... b_0)$ with $x,y$ of the same size
From our preprint On factoring integers of the form $n=(x^D+1)(y^D+1)$ and $n=(x^D+a_{D-2}x^{D-2}+\cdots+ a_0) (y^D+b_{D-2}y^{D-2}+ \cdots+b_0)$ with $x,y$ of the same size.
We got plausible ...
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Prime Inheritance and Prime-Generating Subsequence Trees in Class Number 1 Quadratic Polynomials [closed]
This question is inspired by the classical behavior of Euler’s polynomial
$$
\mathbf{f(x) = x^2 - x + 41},
$$
which is well-known for producing prime values for integer inputs $x = 0$ to $39$, and is ...
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Prime inheritance in class number 1 quadratic polynomials
This conjecture is based on computational exploration of quadratic polynomials associated with imaginary quadratic fields of class number one.
Let us define:
• For a polynomial $f(x) \in \mathbb{Z}[...
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Factoring special form numbers via Galois cubic polynomial
There is a nice family of cubic polynomials with galois group C3 described here: https://math.stackexchange.com/questions/4684769/parametric-family-of-cubics-with-galois-group-c-3 (There are some ...
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Exact form of eigenvalues of pentadiagonal Toeplitz matrices
The tridiagonal Toeplitz matrices
$$\begin{pmatrix}
a & b & & \\
c & \ddots & \ddots \\
& \ddots & \ddots & b \\
& & c ...
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Is there a fast way to find median divisors of a number?
Example: 72 has the following divisors:
1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
The median (middle) divisors are 8 and 9.
Provided we already have the prime factors of a number x, what would be an ...
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Conjecture about gcd of Faulhaber's Polynomials
I asked this question on MSE here
Let $ f_n $ be the $ n $-th Faulhaber polynomial. In this question, I observed the following divisibility properties:
If $ n $ is even, then $ f_n $ is divisible by ...
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Unexpected factorization of a polynomial defined recursively
Define polynomials $f_n(q)$ by $f_1(q)=q^2+q$ and
$$ f_{n+1}(q) = f_n(q)(f_n(q)+2^{2^{n-1}-1}),\ n\geq 1. $$
Then for $n\geq 3$, it seems to be the case that
$f_n(q)+2^{2^{n-1}-1}$ factors (over $\...
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Strongly ambiguous ideals
There are various rumours about links between strongly ambiguous ideals and factors of the discriminant of a number field.
Illustrating this question by way of a concrete example, let
$$f = x^7 + 9x^6 ...
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Is there closed form for the factorization of $F(x)$ of special form modulo $N$?
We got a fast probabilistic algorithm for factorizations of univariate
polynomials of special form modulo $N$ and would like to know if it
is known or trivial.
Let $N$ be composite integer with ...
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Factoring a polynomial that looks almost like a product of $(x + y_1)^{d_1}$ and $(x + y_2)^{d_2}$
I have encountered a polynomial that almost looks like a product of two polynomials of the form $(x+y)^d$, but not quite... The exact polynomial looks like this:
$$
\sum_{0 \leq i_1 \leq d_1} \sum_{0 \...
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Factoring integers by counting number of prime factors in number fields
Quite some ago, the following question was asked on this site: How hard is it to compute the number of prime factors of a given integer?
Terence Tao gave a very interesting answer that involves number ...
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Finding divisors close to the cube root of an integer
For a given integer $N = p\cdot q$ with $p \leq q$, if $q$ is close to $N^{1/2}$, say
$$N^{1/2} \leq q \leq N^{1/2} + (4N)^{1/4}$$
then $p,q$ can be recovered very quickly using Fermat's factoring ...
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On bounds for the factors of integers
Let $N=p q$ and let $D=\frac{\log{q}}{\log{p}} > 1$.
Conjecture 1 There exist positive real constant $A$ depending
only on $D$ such that given integer $r$ in the range
$[q-q^{\frac{D-1}{D}},q+q^{\...
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Given $N=pq$, with what complexity we can find integer $a : q-\sqrt{q} < a <q+\sqrt{q}$?
Related to open problem and this question.
Let $N=p q$ be integer with unknown factorization. We are looking for an answer which can be used in factorization algorithm, so factoring $N$ is not allowed....
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What upper bounds are known, for the number of divisors of Mersenne numbers?
Short version. What upper bounds are known, for the number of divisors of Mersenne numbers?
Long version.
Studying the structure of the factors of $M_n = 2^n - 1$ appears to be an active and difficult ...
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On a probabilistic integer factorization algorithm given bounds for one prime factor
We got a probabilistic integer factorization algorithm and experimental evidence with large
integers given bounds for one factor.
Let $D \ge 2$ be real number and let $p,q$ be primes and $N=pq$.
...
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Factoring totient of a prime
Is it any easy to factor $p-1$ when $p$ is a prime compared to general factorization problem?
What about when $2p+1$ is also a prime?
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What is known about vector subspaces of polynomial rings closed under factors?
Let $R$ be a commutative ring. Call a nonempty subset $F$ of $R$ a factroid if it is closed under sums and factors. That is:
If $a,b \in F$, then $a+b \in F$, and
If $a,b \in R$ with $a\in R$ ...
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Evaluating the coprimality in a bivariate polynomial equation
Given a prime $p>2$, let $x$ and $y$ be real numbers such that $x>y>0$ and
$$ \begin{equation} x^p-y^p=(x-y)^p+pxy(x-y)R \tag{1} \label{eq:one} \end{equation} $$
where $R$ is a bivariate ...
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How dense are quotients of smooth numbers?
As usual, call a positive integer $y$-smooth if it has no prime factors greater than $y$. Write $S(x,y)$ for the set of $y$-smooth integers $\leq x$. Write $R(x,y)$ for the set of quotients $\{a/b: a,...
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How dense are (very but not extremely) smooth numbers? Can they be found in most (not very) short intervals?
An integer is said to be $y$-smooth if it has no prime factors $>y$. Let $y$ be "medium sized", meaning $(\log x)^{1+\epsilon} < y < \exp((\log x)^{2/3})$ or so. (Why this range of ...
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Reference Request: Moore--Postnikov tower of the rationalization of a fibration
Two spaces $X$ and $Y$ are said to be rationally homotopy equivalent, written $X \sim_{\mathbb{Q}} Y$, if their rationalizations $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$
are homotopy equivalent. Moreover,...
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On the parity of $(2^{\varphi(n)}-1) \bmod{n^2}$
For odd integer $n$ define the function
$$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$
$J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$.
Integer $n$ is Wieferich number
iff $J(n)=0$ and if $n$ is ...
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Coprime polynomials and polynomial substitution
Let $F$ be a field, and let $P(X_1,\dots,X_m)$, $Q(X_1,\dots,X_m) \in F[X_1,\dots,X_m]$ be two coprime polynomials. Consider $n$ new polynomials $R_1(Y_{1,1},\dots,Y_{1,n}) \in F[Y_{1,1},\dots,Y_{1,n}]...
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Factorizing groups into a product of solvable subgroups
Does every finite group $G$ have a factorization $G=H_1\cdots H_k$ where the $H_i$ for $1\le i\le k$ are solvable subgroups of $G$ and $|G|=|H_1|\cdots |H_k|$ (equivalently, every element of $G$ is ...
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On an integer factoring algorithm based on smooth class number of quadratic fields
We got an algorithm and toy implementation of integer factoring algorithm
based on smooth class number of quadratic fields.
It is close to the elliptic curve factorization method (ECM) and
succeeds if ...
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Could efficient solutions of $x^2+n y^2=A$ be related to integer factorization?
Let $n$ be positive integer with unknown factorization and $A$ integer with known
factorization.
According to pari/gp developers pari can efficiently find all solutions of:
$$x^2+n y^2=A \qquad (1)$$
...
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Fixed $a_p=p+1-\#E(\mathbb{F}_p)$ and $a_p \ne 0$ on an elliptic curve infinitely often for fixed curve over the rationals?
In this and this question we show that if $p=27a^2+27a+7$ is prime, then the order of the elliptic curve
$y^2=x^3+2$ modulo $p$ is either $p$ or $p+2$.
Q1 Can we unconditionally show that the order ...
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$F(x,y)$ absolutely irreducible over the rationals, but reducible modulo infinitely many primes? [duplicate]
In this two page note we give efficient probabilistic algorithm for factoring bivariate
polynomials in composite characteristic assuming the solution is unique
and we would like to test the algorithm ...
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On a efficient algorithm for factoring bivariate polynomials modulo composite modulus assuming the solution is unique
We found and implemented in sage efficient algorithm for factoring
bivariate polynomials modulo composite modulus assuming the solution is unique up to a constant factor.
More formally let $K=\mathbb{...
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Does this sequence ever end?
This may help: A080670 A195265
Define $f(n)$ as this:
Take a number $n$, and split it into its prime composition using $^$ and $×$. Now remove all $^$ and $×$, you get a new number, this is $f(n)$ (...
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Number of distinct exponent patterns in the prime power factorizations of the integers 1,2,...,n
Let $n=p_1^{a_1}\cdots p_k^{a_k}$ be the prime power factorization of
the positive integer $n$, with $p_1<\cdots<p_k$ and $a_i>0$. Define
$\kappa(n)=(a_1,\dots,a_k)$, the composition type of $...
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Effective Erdős–Kac theorem
I have some number $N$ and some integer $k>0$. I want to know what fraction of numbers up to $N$ have more than $k$ prime factors. (In my application, with repetition, but the $\omega$ version is ...
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Adjoining new factors for primes in UFDs
It is well-known that if we pass from a UFD to a new ring where we have factored one of the primes, it does not need to stay a UFD. The classic example is passing from $\mathbb{Z}$ to $\mathbb{Z}[\...
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Primality testing by reversible computation using the prime number theorem
Suppose we want to build a primality testing algorithm for the numbers limited to the set $A =\{1, ..., 2^n\}$ and $n$ is reasonably large. The prime-number theorem tells us that there are ...
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Coefficients of 0,1-polynomials factorization
Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$.
Let $Q_{1}(x) \cdot Q_{2}(x) \cdots Q_{m}(x)$ - polynomial factorization (over integers) of $P_{n}$.
...
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Reference Request: Factorization method for polynomials whose maximum absolute value of coefficient is 1
So, today I came up with a method for factoring polynomials whose coefficients are either $-1$ or $1.$
Let me explain with examples.
Example No. 1. Factorize $P(x)=x^8+x^7+1$
Solution. It is known ...
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Simple question about 0,1-polynomials
Being interested in these polynomials, would like to clarify one small observation.
Let $n\in\mathbb{N}$ and $P_{n}$ is $0,1$-polynomial whose coefficients are binary digits of $n$.
Let $n$ has prime ...
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Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation
Happy New Year, MO community!
We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem.
PROBLEM
...
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Has anyone studied factoring as a CO-product?
In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors.
I want to explore the "Co-ness" of this....
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Uniqueness of sum of squares representation
Given a polynomial $f(x) \in \mathbb{R}[x] = \mathbb{R}[x_{1},\dots,x_{n}]$. We say $f(x)$ is sum of squares(SOS) if there are polynomials, $p_{1},\dots,p_{k}$ such that $f = p_{1}^{2} + \dots+p_{k}^{...
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Closed formula for number of ones in a proper factor tree
Edit [2023 Dec 7]: One of my specific wonders, along with that of students, is around when a recursive formula might have – or be expected to have – an explicit or closed formula. What is the ...
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How to do LU factorization efficiently based on the factorized result added with a low-rank matrix?
Suppose a square $n\times n$, dense matrix $A^{\text{old}}$ has been factorized into $L^{\text{old}}$ and $U^{\text{old}}$ components by performing a LU decomposition $A^{\text{old}} = L^{\text{old}}U^...
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A doubt regarding the extended form of the Weierstrass factorization theorem
I want to represent $\sin(x)-\dfrac{1}{\sqrt{2}}$ as a product of it's zeroes
According to the Weierstrass factorization theorem, the sine function can be represented as a product of its factors:
$$\...
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Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
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